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Combining subspace codes

The work of the first, third and fourth author was supported by the Italian National Group for Algebraic and Geometric Structures and their Applications (GNSAGA–INdAM)

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  • In the context of constant-dimension subspace codes, an important problem is to determine the largest possible size $ A_q(n, d; k) $ of codes whose codewords are $ k $-subspaces of $ {\mathbb F}_q^n $ with minimum subspace distance $ d $. Here in order to obtain improved constructions, we investigate several approaches to combine subspace codes. This allow us to present improvements on the lower bounds for constant-dimension subspace codes for many parameters, including $ A_q(10, 4; 5) $, $ A_q(12, 4; 4) $, $ A_q(12, 6, 6) $ and $ A_q(16, 4; 4) $.

    Mathematics Subject Classification: Primary: 51E20; Secondary: 05B25, 94B65.

    Citation:

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  • [1] J. André, Über nicht-Desarguessche Ebenen mit transitiver Translationsgruppe, Math. Z., 60 (1954), 156-186.  doi: 10.1007/BF01187370.
    [2] A. Beutelspacher, On parallelisms in finite projective spaces, Geometriae Dedicata, 3 (1974), 35-40.  doi: 10.1007/BF00181359.
    [3] M. BraunP. R. J. Östergård and A. Wassermann, New lower bounds for binary constant-dimension subspace codes, Exp. Math., 27 (2018), 179-183.  doi: 10.1080/10586458.2016.1239145.
    [4] H. ChenX. HeJ. Weng and L. Xu, New constructions of subspace codes using subsets of MRD codes in several blocks, IEEE Trans. Inform. Theory, 66 (2020), 5317-5321.  doi: 10.1109/TIT.2020.2975776.
    [5] A. Cossidente, G. Marino and F. Pavese, Subspace code constructions, Ric. di Mat., (2020). to appear. doi: 10.1007/s11587-020-00521-9.
    [6] A. Cossidente and F. Pavese, On subspace codes, Des. Codes Cryptogr., 78 (2016), 527-531.  doi: 10.1007/s10623-014-0018-6.
    [7] A. Cossidente and F. Pavese, Veronese subspace codes, Des. Codes Cryptogr., 81 (2016), 445-457.  doi: 10.1007/s10623-015-0166-3.
    [8] A. Cossidente and F. Pavese, Subspace codes in ${\rm{PG(2N - 1, Q)}}$, Combinatorica, 37 (2017), 1073-1095.  doi: 10.1007/s00493-016-3354-5.
    [9] A. Cossidente, F. Pavese and L. Storme, Geometrical aspects of subspace codes, in Network Coding and Subspace Designs, Springer, Cham, (2018), 107-129.
    [10] P. Delsarte, Bilinear forms over a finite field, with applications to coding theory, J. Combin. Theory Ser. A, 25 (1978), 226-241.  doi: 10.1016/0097-3165(78)90015-8.
    [11] R. H. F. Denniston, Some packings of projective spaces, Atti Accad. Naz. Lincei Rend. Cl. Sci. Fis. Mat. Nat., 52 (1972), 36-40. 
    [12] T. Etzion and N. Silberstein, Error-correcting codes in projective spaces via rank-metric codes and Ferrers diagrams, IEEE Trans. Inform. Theory, 55 (2009), 2909-2919.  doi: 10.1109/TIT.2009.2021376.
    [13] T. Etzion and N. Silberstein, Codes and designs related to lifted MRD codes, IEEE Trans. Inform. Theory, 59 (2013), 1004-1017.  doi: 10.1109/TIT.2012.2220119.
    [14] P. Frankl and V. Rödl, Near perfect coverings in graphs and hypergraphs, European J. Combin., 6 (1985), 317-326.  doi: 10.1016/S0195-6698(85)80045-7.
    [15] M. Gadouleau and Z. Yan, Constant-rank codes and their connection to constant-dimension codes, IEEE Trans. Inform. Theory, 56 (2010), 3207-3216.  doi: 10.1109/TIT.2010.2048447.
    [16] H. Gluesing-Luerssen and C. Troha, Construction of subspace codes through linkage, Adv. Math. Commun., 10 (2016), 525-540.  doi: 10.3934/amc.2016023.
    [17] X. He, Construction of constant dimension code from two parallel versions of linkage construction, IEEE Communi. Lett., 24 (2020), 2392-2395. 
    [18] X. He and Y. Chen, Construction of constant dimension codes from several parallel lifted MRD code, preprint, arXiv: 1911.00154.
    [19] D. Heinlein, New LMRD code bounds for constant dimension codes and improved constructions, IEEE Trans. Inform. Theory, 65 (2019), 4822-4830.  doi: 10.1109/TIT.2019.2905002.
    [20] D. Heinlein, Generalized linkage construction for constant-dimension codes, IEEE Trans. Inform. Theory, 67 (2020), 705-715. 
    [21] D. HeinleinT. HonoldM. KiermaierS. Kurz and A. Wassermann, Classifying optimal binary subspace codes of length $8$, constant dimension $4$ and minimum distance $6$, Des. Codes Cryptogr., 87 (2019), 375-391.  doi: 10.1007/s10623-018-0544-8.
    [22] D. Heinlein, M. Kiermaier, S. Kurz and A. Wassermann, Tables of subspace codes, preprint, arXiv: 1601.02864.
    [23] D. Heinlein and S. Kurz, Asymptotic bounds for the sizes of constant dimension codes and an improved lower bound, in Coding Theory and Applications : 5th International Castle Meeting, ICMCTA 2017, Vihula, Estonia, August 28-31, 2017, Proceedings, vol. 10495 of Lecture Notes in Computer Science, Springer International Publishing, Cham, (2017), 163-191. doi: 10.1007/978-3-319-66278-7_15.
    [24] D. Heinlein and S. Kurz, Coset construction for subspace codes, IEEE Trans. Inform. Theory, 63 (2017), 7651-7660.  doi: 10.1109/TIT.2017.2753822.
    [25] T. Honold and M. Kiermaier, On putative $q$-analogues of the Fano plane and related combinatorial structures, in Dynamical Systems, Number Theory and Applications: A Festschrift in Honor of Armin Leutbecher's 80th Birthday, World Scientific, (2016), 141-175.
    [26] T. Honold, M. Kiermaier and S. Kurz, Optimal binary subspace codes of length $6$, constant dimension $3$ and minimum subspace distance $4$, Topics in Finite Fields, Contemp. Math., Amer. Math. Soc., Providence, RI, 632 (2015), 157-176. doi: 10.1090/conm/632/12627.
    [27] T. Honold, M. Kiermaier and S. Kurz, Partial spreads and vector space partitions, in Network Coding and Subspace Designs, Springer, Cham, (2018), 131-170.
    [28] A.-L. Horlemann-Trautmann and J. Rosenthal, Constructions of constant dimension codes, in Network Coding and Subspace Designs, Springer, Cham, (2018), 25-42.
    [29] M. Kiermaier and S. Kurz, On the lengths of divisible codes, IEEE Trans. Inform. Theory, 66 (2020), 4051-4060.  doi: 10.1109/TIT.2020.2968832.
    [30] R. Koetter and F. R. Kschischang, Coding for errors and erasures in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3579-3591.  doi: 10.1109/TIT.2008.926449.
    [31] A. Kohnert and S. Kurz, Construction of large constant dimension codes with a prescribed minimum distance, in Mathematical methods in computer science, Springer, (2008), 31-42. doi: 10.1007/978-3-540-89994-5_4.
    [32] S. Kurz, Packing vector spaces into vector spaces, Australas. J. Combin., 68 (2017), 122-130. 
    [33] S. Kurz, A note on the linkage construction for constant dimension codes, preprint, arXiv: 1906.09780.
    [34] S. Kurz, Subspaces intersecting in at most a point, Des. Codes Cryptogr., 88 (2020), 595-599.  doi: 10.1007/s10623-019-00699-6.
    [35] S. LiuY. Chang and T. Feng, Parallel multilevel construction for constant dimension codes, IEEE Trans. Inform. Theory, 66 (2020), 6884-6897.  doi: 10.1109/TIT.2020.3004315.
    [36] E. L. Năstase and P. A. Sissokho, The maximum size of a partial spread in a finite projective space, J. Combin. Theory Ser. A, 152 (2017), 353-362.  doi: 10.1016/j.jcta.2017.06.012.
    [37] B. Segre, Teoria di Galois, fibrazioni proiettive e geometrie non desarguesiane, Ann. Mat. Pura Appl., 64 (1964), 1-76.  doi: 10.1007/BF02410047.
    [38] J. Sheekey, MRD codes: Constructions and connections, in Combinatorics and Finite Fields: Difference Sets, Polynomials, Pseudorandomness and Applications, vol. 23 of Radon Series on Computational and Applied Mathematics, De Gruyter, Berlin, 2019. doi: 10.1515/9783110642094-013.
    [39] N. Silberstein and A.-L. Trautmann, Subspace codes based on graph matchings, Ferrers diagrams, and pending blocks, IEEE Trans. Inform. Theory, 61 (2015), 3937-3953.  doi: 10.1109/TIT.2015.2435743.
    [40] D. SilvaF. R. Kschischang and R. Köetter, A rank-metric approach to error control in random network coding, IEEE Trans. Inform. Theory, 54 (2008), 3951-3967.  doi: 10.1109/TIT.2008.928291.
    [41] H. WangC. Xing and R. Safavi-Naini, Linear authentication codes: Bounds and constructions, IEEE Trans. Inform. Theory, 49 (2003), 866-872.  doi: 10.1109/TIT.2003.809567.
    [42] S.-T. Xia and F.-W. Fu, Johnson type bounds on constant dimension codes, Des. Codes Cryptogr., 50 (2009), 163-172.  doi: 10.1007/s10623-008-9221-7.
    [43] L. Xu and H. Chen, New constant-dimension subspace codes from maximum rank distance codes, IEEE Trans. Inform. Theory, 64 (2018), 6315-6319.  doi: 10.1109/TIT.2018.2839596.
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